Problem 8 - A Gentle Introduction to the Art of Mathematics

Question

In proving the product rule in Calculus using the definition of the derivative, we might start our proof with: $$\begin{array}{c} \frac{\mathrm{d}}{\mathrm{d} x}(f(x) \cdot g(x)) \\ =\lim _{h \longrightarrow 0} \frac{f(x+h) \cdot g(x+h)-f(x) \cdot g(x)}{h} \end{array}$$ The last two lines of our proof should be: $$\begin{array}{c} =\lim _{h \longrightarrow 0} \frac{f(x+h)-f(x)}{h} \cdot g(x)+f(x) \cdot \lim _{h \longrightarrow 0} \frac{g(x+h)-g(x)}{h} \\ =\frac{\mathrm{d}}{\mathrm{d} x}(f(x)) \cdot g(x)+f(x) \cdot \frac{\mathrm{d}}{\mathrm{d} x}(g(x)) \end{array}$$ Fill in the rest of the proof.

Step-by-Step Solution

Answer

Using the limit definition, distribute and simplify to get \( \frac{ \mathrm{d} }{ \mathrm{d} x } (f(x) g(x)) = f(x) \cdot \frac{\mathrm{d}}{\mathrm{d} x}(g(x)) + g(x) \cdot \frac{\mathrm{d}}{\mathrm{d} x}(f(x)) \).

Step 1: Restate the problem with the definition of the derivative

Start the proof using the limit definition of the derivative for the product of two functions. The expression is: \[ \frac{\mathrm{d}}{\mathrm{d} x}(f(x) \cdot g(x)) = \lim\_{h \rightarrow 0} \frac{f(x+h) \cdot g(x+h) - f(x) \cdot g(x)}{h} \]

Step 2: Add and subtract a term

To simplify the limit, add and subtract \( f(x+h) \cdot g(x) \) within the numerator: \[ \lim\_{h \rightarrow 0} \frac{f(x+h) \cdot g(x+h) - f(x+h) \cdot g(x) + f(x+h) \cdot g(x) - f(x) \cdot g(x)}{h} \]

Step 3: Split the limit into two parts

Use the property of limits to split the expression into two limits: \[ \lim\_{h \rightarrow 0} \frac{ f(x + h) \cdot [ g(x + h) - g(x) ] + [ f(x + h) - f(x) ] \cdot g(x) }{ h } \]

Step 4: Distribute the limit

Distribute the limit across the numerator by splitting it into two separate limits: \[ \lim\_{h \rightarrow 0} \left[ f(x + h) \cdot \frac{ g(x + h) - g(x) }{ h } + \frac{ f(x + h) - f(x) }{ h } \cdot g(x) \right] \]

Step 5: Take the limit

Evaluate each term separately as \( h \rightarrow 0 \): \[ = \lim\_{h \rightarrow 0} f(x + h) \cdot \lim\_{h \rightarrow 0} \frac{ g(x + h) - g(x) }{ h } + \lim\_{h \rightarrow 0} \frac{ f(x + h) - f(x) }{ h } \cdot \lim\_{h \rightarrow 0} g(x) \] Recall that \( \lim\_{h \rightarrow 0} f(x + h) = f(x) \) and \( \lim\_{h \rightarrow 0} g(x) = g(x) \): \[ = f(x) \cdot \frac{ \mathrm{d} }{ \mathrm{d} x } g(x) + \frac{ \mathrm{d} }{ \mathrm{d} x } f(x) \cdot g(x) \]

Key Concepts

definition of derivative

Let's start with the basic definition of a derivative. The derivative of a function measures how the function's value changes as its input changes. Formally, for a function \( f(x) \), the derivative is defined as:
\( f'(x) = \frac{d}{dx} f(x) = \frac{\text{d} f(x)}{\text{d} x} = \frac{f(x + h) - f(x)}{h}, \text{as} \, h \rightarrow 0 \).
This formula means we see how much \( f(x) \) changes when \( x \) changes by a small amount \( h \).
This is also known as the limit definition of the derivative.

limit properties

Limits are foundational in calculus. They help us understand behavior as values get closer and closer to a point.
Here are a few important limit properties:

• Sum Rule: \( \text{lim}_{x \rightarrow a} [f(x) + g(x)] = \text{lim}_{x \rightarrow a} f(x) + \text{lim}_{x \rightarrow a} g(x) \)
• Product Rule: \( \text{lim}_{x \rightarrow a} [f(x) \times g(x)] = \text{lim}_{x \rightarrow a} f(x) \times \text{lim}_{x \rightarrow a} g(x) \)
• Quotient Rule: If \( g(a) eq 0 \), \( \text{lim}_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\text{lim}_{x \rightarrow a} f(x)}{\text{lim}_{x \rightarrow a} g(x)} \)

calculus proof steps

Proving the product rule in calculus can be broken down into several important steps:

• Step 1: Restate the problem using the definition of the derivative.
  This gives us: \( \frac{\text{d}}{\text{d} x}(f(x) \times g(x)) = \text{lim}_{h \rightarrow 0} \frac{f(x+h) \times g(x+h) - f(x) \times g(x)}{h} \).
• Step 2: Add and subtract a common term within the numerator.
  This term is typically something like \( f(x+h) \times g(x) \).
  This simplifies our expression: \( \text{lim}_{h \rightarrow 0} \frac{f(x+h) \times g(x+h) - f(x+h) \times g(x) + f(x+h) \times g(x) - f(x) \times g(x)}{h} \).
• Step 3: Split the limit using our earlier limit properties.
  \( \text{lim}_{h \rightarrow 0} [ f(x + h) \times \frac{g(x + h) - g(x)}{h} ] + \text{lim}_{h \rightarrow 0} [ \frac{f(x + h) - f(x)}{h} \times g(x) ] \).
• Step 4: Distribute the limit across.
  \( \text{lim}_{h \rightarrow 0} [ f(x + h) \times \text{lim}_{h \rightarrow 0} \frac{g(x + h) - g(x)}{h} ] + \text{lim}_{h \rightarrow 0}[ \frac{f(x + h) - f(x)}{h} \times g(x) ] \).
• Step 5: Evaluate the limit expressions.
  Recognize that \( \text{lim}_{h \rightarrow 0} f(x + h) = f(x) \) and \( \text{lim}_{h \rightarrow 0} g(x + h) = g(x) \).
  This gives us: \( f(x) \times \frac{ \text{d} }{ \text{d} x } g(x) + \frac{ \text{d} }{ \text{d} x } f(x) \times g(x) \).

derivative rules

Derivative rules can simplify finding derivatives. One important rule is the product rule, useful for differentiating products of functions.
The product rule states:
\( \frac{\text{d}}{\text{d} x} [ f(x) \times g(x) ] = f'(x) \times g(x) + f(x) \times g'(x) \).
Another essential rule is the sum rule, which says:
\( \frac{\text{d}}{\text{d} x} [ f(x) + g(x) ] = \frac{\text{d}}{\text{d} x} f(x) + \frac{\text{d}}{\text{d} x} g(x) \).
Quicker derivative rules, including the power rule, quotient rule, and chain rule, also exist to help with more complex functions.